The problem turned out to be a DNS-related issue. By issuing a few ping commands from the access point, we would see that it was able to get the needed connectivity, but were unable to resolve “cisco-capwap-controller”.

Now, this DNS option had been missing since forever, so my guess was that something changed within the DNS-server itself. I would imagine that previously the DNS-server automatically assumed omit.local as DNS suffix if missing from the DNS-request.
Anyway, hopefully this will assist someone who might be experiencing the same issue.